This coefficient is related to the \(\hat{E}^3/j^4\)  term of the periastron advance \(K^{\rm loc,h}(\hat{E},j)\) .
The potential contributions to this quantity are of \(O(G_N^4)\)  and have been first obtained in
[1]}. They have recently been confirmed by the \(O(G_N^4)\)  post–Minkowskian
results of [2]}, cf. also [3]}, [4]}. The \(O(\nu ^2 \pi ^2)\) 
contribution to this term is implied
by \(q_{44}\) , since \(q_{82}\)  and \(q_{63}\)  are purely rational in \(\nu \) . We also would like to mention that
the \(\pi ^2\)  terms of \(\bar{d}_5\)  and \(a_6\) , as determined using the scattering angle in [5]}, using
the potential terms given in [1]}, are equivalently obtained by using Eqs. (REF ,
REF ), which has been presented before in Ref. [1]}.
